Oscillation of Even-Order Neutral Delay Differential Equations
© Tongxing Li et al. 2010
Received: 28 November 2009
Accepted: 29 March 2010
Published: 19 May 2010
By using Riccati transformation technique, we will establish some new oscillation criteria for the even order neutral delay differential equations , , where is even, , , and . These oscillation criteria, at least in some sense, complement and improve those of Zafer (1998) and Zhang et al. (2010). An example is considered to illustrate the main results.
where is even
In what follows we assume that
(I2) , , , , , , where is a constant,
(I3) and for is a constant.
Neutral differential equations find numerous applications in natural science and technology. For instance, they are frequently used for the study of distributed networks containing lossless transmission lines; see Hale .
To the best of our knowledge, the above oscillation results cannot be applied when and it seems to have few oscillation results for (1.1) when
Motivated by Liu and Bai , we will further the investigation and offer some more general new oscillation criteria for (1.1), by employing a class of function and operator The method used in this paper is different from .
for and has the partial derivative on such that is locally integrable with respect to in
By choosing the special function it is possible to derive several oscillation criteria for a wide range of differential equations.
2. Main Results
In this section, we give some new oscillation criteria for (1.1). In order to prove our theorems we will need the following lemmas.
Lemma 2.1 (see ).
Let If is eventually of one sign for all large say then there exist a and an integer with even for or odd for such that implies that for and implies that for
Lemma 2.2 (see ).
Lemma 2.3 (see ).
Assume that .
where the operator is defined by (1.9), and is defined by (1.10). Then every solution of (1.1) is oscillatory.
Let be a nonoscillatory solution of (1.1). Then there exists such that for all Without loss of generality, we assume that for all
which contradicts (2.3). This completes the proof.
We get the following new result.
where is defined as in Theorem 2.4, the operator is defined by (1.9), and is defined by (1.10). Then every solution of (2.20) is oscillatory.
Let be a nonoscillatory solution of (2.20). Then there exists such that for all
Then The rest of the proof is similar to that of the proof of Theorem 2.4, hence we omit the details.
By Theorem 2.4 (or Theorem 2.5) we can obtain the oscillation criterion for (1.1) (or (2.20)); the details are left to the reader.
For an application, we give the following example to illustrate the main results.
By Theorem 2.5, let one has (2.21). Hence, every solution of (2.27) oscillates. For example, is an oscillatory solution of (2.27).
The recent results cannot be applied in (1.1) and (2.20) when for Therefore, our results are new.
It would be interesting to find another method to study (1.1) and (2.20) when , or for
It would be more interesting to find another method to study (1.1) when is odd.
The authors sincerely thank the reviewers for their valuable suggestions and useful comments that have led to the present improved version of the original manuscript. This research is supported by the Natural Science Foundation of China (60774004, 60904024), China Postdoctoral Science Foundation funded Project (20080441126, 200902564), and Shandong Postdoctoral funded project (200802018) by the Natural Science Foundation of Shandong (Y2008A28, ZR2009AL003), and also by University of Jinan Research Funds for Doctors (B0621, XBS0843).
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